Pattern formation mechanisms in motility mutants of Myxococcus

Pattern formation mechanisms in motility mutants of Myxococcus xanthus
Jörn Starruß∗ ,1, 2 Fernando Peruani † ‡ ,1, 3 Vladimir Jakovljevic,4
Lotte Søgaard-Andersen,4 Andreas Deutsch,2 and Markus Bär5
1
These authors have contributed equally to this work.
Center for Information Services and High Performance Computing (ZIH),
Technische Universität Dresden, Zellescher Weg 12, D-01069 Dresden, Germany
3
Laboratoire J.A. Dieudonné, Université de Nice Sophia Antipolis,
UMR 7351 CNRS , Parc Valrose, F-06108 Nice Cedex 02, France
4
Max Planck Institute for Terrestrial Microbiology,
Karl-von-Frisch Straße 10, D-35043 Marburg, Germany
5
Physikalisch-Technische Bundesanstalt, Abbestraße 2-12, 10587 Berlin, Germany
(Dated: September 12, 2012)
2
Formation of spatial patterns of cells is a recurring theme in biology and often depends on regulated cell motility. Motility of the rod-shaped cells of the bacterium Myxococcus xanthus depends
on two motility machineries, type IV pili (giving rise S-motility) and the gliding motility apparatus
(giving rise to A-motility). Cell motility is regulated by occasional reversals. Moving M. xanthus
cells can organize into spreading colonies or spore-filled fruiting bodies depending on their nutritional
status. To ultimately understand these two pattern formation processes and the contributions by the
two motility machineries, as well as the cell reversal machinery, we analyze spatial self-organization
in three M. xanthus strains: i) a mutant that moves unidirectionally without reversing by the Amotility system only, ii) a unidirectional mutant that is also equipped with the S-motility system,
and iii) the wild-type that, in addition to the two motility systems, occasionally reverses its direction
of movement. The mutant moving by means of the A-engine illustrates that collective motion in the
form of large moving clusters can arise in gliding bacteria due to steric interactions of the rod-shaped
cells, without the need of invoking any biochemical signal regulation. The two-engine strain mutant
reveals that the same phenomenon emerges when both motility systems are present, and as long
as cells exhibit unidirectional motion only. From the study of these two strains, we conclude that
unidirectional cell motion induces the formation of large moving clusters at low and intermediate
densities, while it results into vortex formation at very high densities. These findings are consistent
with what is known from self-propelled rod models which strongly suggests that the combined effect
of self-propulsion and volume exclusion interactions is the pattern formation mechanism leading to
the observed phenomena. On the other hand, we learn that when cells occasionally reverse their
moving direction, as observed in the wild-type, cells form small but strongly elongated clusters and
self-organize into a mesh-like structure at high enough densities. These results have been obtained
from a careful analysis of the cluster statistics of ensembles of cells, and analyzed on the light of a
coagulation Smoluchowski equation with fragmentation.
PACS numbers:
Formation of patterns of spatially organized cells is a
recurring theme in biology. These processes often depend
on regulation of cell motility. For instance, in metazoans it provides the basis for organ formation during
embryogenesis and, in single-celled eukaryotes such as
Dictyostelium discoideum, it is essential for the formation of fruiting bodies. In bacteria, regulated cell motility is essential for the colonization of diverse habitats
as well as for the formation of multicellular structures
such as biofilms and fruiting bodies. These pattern formation processes can be self-organized. For example,
Dictyostelium discoideum [1, 2] regulates cell aggregation
and multicellular organization by secreting and sensing
the diffusive signal cAMP. In Myxococcus xanthus, on the
∗ These
authors have contributed equally to this work.
authors have contributed equally to this work.
‡ Corresponding author: [email protected]
† These
other hand, rippling patterns and the highly complex cellular reorganization leading to fruiting body formation
are controlled by a non-diffusing signal, the C-signal [3].
Interestingly, collective effects and self-organization can
also occur, to a certain extent, in the absence of an
explicit signaling mechanism. For instance, hydrodynamic interactions can induce large-scale coherent motion of swimming cells, as recently observed in Bacillus
subtilis[4, 5], and a density-dependent diffusivity can lead
to aggregation patterns as recently suggested to occur in
Escherichia coli and Salmonella typhimurium [6].
M. xanthus is a gliding bacterium that has been used as
a model system to study pattern formation [14], bacterial
social behaviour [15], and motility [16]. The rod-shaped
cells of the bacterium M. xanthus move on surfaces in the
direction of their long axis using two motility machineries, type IV pili, which requires cell-to-cell contact for its
activity beacuse it is stimulated by exopolysaccharides
on neighboruing cells [17] (giving rise S-motility), and
the gliding motility apparatus that allows cells to move
2
FIG. 1: Pattern formation at various packing fractions η, 6 hs after spotting a drop of bacterial suspension on an agar surface.
First row corresponds to the non-reversing A+ S− Frz− mutant that moves only by means of the A-motility system (these three
panels have been taken from [7]). Second row corresponds to the non-reversing A+ S+ Frz− mutant that moves with both the
A- and S-motility system. At high cell densities, the mutants A+ S− Frz− and A+ S+ Frz− form large moving clusters that turn
into vortices at sufficiently high packing fractions. Third row corresponds to the wild-type A+ S+ Frz+ strain that moves with
both the A- and S-motility systems and cells are able to reverse their moving direction. The A+ S+ Frz+ mutant self-organizes
into a mesh-like structure at high density.
in isolation [18] (giving rise to A-motility). Force generation by the A-motility system has been suggested to rely
on either slime secretion from the lagging pole [22], or on
focal adhesion complexes distributed along the cell [23].
Cells occasionally reverse their gliding direction with an
average frequency of about once per 10 min and the reversal frequency is controlled by the Frz chemosensory
system [19]. In the presence of nutrients, M. xanthus
cells form coordinately spreading colonies. Upon depletion of nutrients M. xanthus cells initiate a complex developmental program that culminates in the formation
of spore-filled fruiting bodies. Both motility systems as
well as reversals are required for the two cellular patterns
to form, i.e., spreading colonies and fruiting bodies. It is
currently not known how the reversal frequency is regulated except that cell-cell contacts may induce C-signal
exchange which is supposed to stimulate reversals during rippling and to inhibit reversals during aggregation.
During fruiting body formation the reversal frequency
decreases up to a point where cell movements become
nearly unidirectional [20] and cells start to display collective motion with the formation of large clusters in which
cells are aligned in parallel making side-to-side as well as
head-to-tail contacts and move in the same direction [21].
Eventually cells start to aggregate. Aggregation centers
often resemble at their initial phase a cell vortex.
Here, we aim at understanding myxobacterial pattern
formation processes, particularly the contributions by the
two motility machineries as well as the cell reversal machinery to the spatial organization of the cells. We study
the role of steric interactions, cell adhesion, and reversal
frequency on the collective dynamics. The question for
us is not “why” cells exhibit a given collective behavior
but “how” they do it. In order to identify the role of
the two motility machineries and cell reversal machinery,
we follow a bottom-up strategy by looking at the collective dynamics of different mutants of increasing complexity. We analyze three M. xanthus strains: i) a mutant
that moves unidirectionally without reversing by the Amotility system only – mutant that has been previously
studied by us in [7] –, ii) a unidirectional mutant that
is also equipped with the S-motility system, and iii) the
wild-type that, in addition to the two motility systems,
occasionally reverses its direction of movement. We char-
3
acterize the macroscopic patterns mainly through the
cluster statistics, in particular in terms of cluster size and
shape. We observe that the mutant moving by means of
the A-engine only displays collective motion in the form
of large moving clusters. The study of its cluster size distribution reveals that above a given density, clusters can
be arbitrary large [7]. Here, we show in addition that
there is a non trivial scaling of cluster perimeter with
cluster size which indicates that the clustering process
is neither (fully) random nor as in (equilibrium) liquidvapor drops [10]. We also find that at high densities the
collective dynamics changes and cells organize into vortices. The study of the two-engine strain mutant reveals
the same phenomenology for these bacteria: collective
motion in the form of large moving clusters, a critical
density above which cluster can be arbitrarily large, a
non-trivial scaling of cluster perimeter with cluster size,
and vortex formation at high densities. From the comparison of these two strains, we conclude that unidirectional cell motion induces the formation of large moving
clusters at low and intermediate densities, while it results
into vortex formation at very high densities, see first and
second row of panels in Fig. 1. Interestingly, similar collective dynamics has been observed in self-propelled rod
models [11], a fact that strongly suggests that the combined effect of self-propulsion and volume exclusion interactions is the pattern formation mechanism leading to
the observed phenomena.
The study of wild-type cells indicates that cell reversal
weakens clustering. Wild-type cells exhibit exponential
cluster size distributions at low and intermediate densities, while the scaling of the cluster perimeter with cluster size indicates that clusters are strongly elongated. At
high densities, we find that reversing wild-type cells selforganize into a mesh-like structure, see bottom row in
Fig. 1.
Wild-type cells, as commented above, exhibit a large
variety of self-organized patterns depending on the environmental condition. Our results suggest that by only
switching on and off the reversal, cells can modify dramatically their collective behavior, with the suppression
of cell reversal leading to collective motion in the form
of moving clusters and vortex formation at high densities. This observation is consistent with the observed decrease in reversal frequency in the wild-type upon nutrient depletion, which is followed by the formation of large
moving clusters and aggregation of cells. Our findings
indicate that these two processes can result from simple steric interactions of the (non-reversing) rod-shaped
cells, without the need of invoking any biochemical signal
regulation.
The paper is organized as follows. In Sec.I.a, we focus
on the spatial self-organization of purely A-motile cells in
the absence of cell reversals. The effects induced by the
S-motility engine, which include increased cell adhesion,
are studied in Sec.I.b, while those due to cell reversals in
Sec.I.c. In Sec.II we discuss which collective effects are
expected in self-propelled rod models, and interpret the
cluster statistics results observed in the experiments in
the light of a simple cluster formation theory. We summarize all the results in Sec. III, where we also discuss
the implications of the reported findings.
I.
A.
CLUSTER STATISTICS
A-motile non-reversing cells
We start out with the simple mutant A+ S− Frz− that
only moves by means of the A-motility system and which
is unable to reverse due to an insertion in the frz gene
cluster (see Material and Methods for more details about
how the strain was generated). This mutant is unable to
assemble type IV pili due to deletion of the pilA gene,
which encodes the type IV pili subunit, and therefore
the S-motility system is non-functional in this mutant.
This mutant exhibits relatively weak cell-cell adhesion
due to the lack of type IV pili and the reduced accumulation of exopolysaccharides. This mutant is labeled
A+ S− Frz− to indicate that A-motility engine is on, the
S-motility engine is off and the Frz system, i.e., cell reversal is dramatically reduced. Control experiments showed
that these mutants have a reversal period 100 min
whereas the isogenic Frz+ strain reversed with a mean
reversal period of ∼ 10 min. In [7], we showed that this
mutant exhibits a transition to a collective motion phase
at high enough densities by analyzing the dependency of
cluster size distribution with the packing fraction. Here,
we characterize in addition the cluster shape, and show
that at densities higher than the one studied in [7], giant
clusters turn into vortices.
Experiments were performed by spotting a drop of cell
suspension of the desired density on an agar surface to
subsequently monitor the evolution of cell arrangements
by taking snapshots of the bacterial colony every 30 min
for a total of 8 hrs. Experiments with cells gliding in
isolation indicate an average velocity of v = 3.10 ± 0.35
µm/min, an average width of about W = 0.7 µm and an
average length of L = 6.3 µm. This results in a mean
aspect ratio of κ = L/W = 8.9 ± 1.95 and a cell covering
an average area a = 4.4 µm2 .
We found that under these conditions cells organized
over time into moving clusters. Time-lapse recordings
showed that collisions of cells lead to effective alignment
(Fig. 2a). When the interaction is such that cells end
up parallel to each other and move in the same direction, they migrate together for a long time (typically
> 15 min). Eventually, successive collisions allow a small
initial cluster to grow in size, Fig. 2. In the individual clusters, cells are aligned in parallel to each other
and arranged in a head-to-tail manner, as previously described [24]. In a cluster, cells move in the same direction. Cluster-cluster collision typically leads to cluster
fusion, whereas splitting and break-up of clusters rarely
occur. On the other hand, individual cells on the border
of a cluster often spontaneously escape from the cluster.
4
FIG. 2: (a) Collisions among M. xanthus lead to an effective (local) alignment. (b) and (c) show that a local alignment leads
to formation of moving clusters; arrows indicate the cluster moving direction. Time interval between (b) and (c) is 15 min,
snapshots correspond to A+ S− Frz− cells at packing fraction η = 0.11. Panels taken from [7].
These two effects, cluster growth due to cluster-cluster
collision and cluster shrinkage, mainly due to cells escaping from the cluster boundary, compete and give rise to
a characteristic cluster size distribution (CSD).
The CSD - p(m, t) - indicates the probability of a bacterium to be in a cluster of size m at time t. Note that
along the text, the term CSD always refers to this definition. Often times the cluster size distribution is alternatively determined as the number nm (t) of clusters of size
m at time t. There is a simple relation between these two
definitions: pm (t) ∝ m nm (t). In experiments we have
observed that the CSD mainly depends on the packing
fraction η, where η = ρ a, with ρ the (two-dimensional)
cell density and a the average covering area of a bacterium given above. Hence, for all snapshots first the
packing fraction was determined. Then, images with
similar packing fraction η were compared and the CSD
was reconstructed by determining the CSD for all images
within a finite interval of the packing fraction. Very importantly, we find that the CSD p(m, t) reaches a steady
state p(m) after some transient time, as shown in Fig. 3.
We conclude that the clustering process evolves towards
a dynamic equilibrium, where the process of formation
of cell clusters of a given size is balanced by events in
which clusters of this size disappear by either fusing with
other clusters or by loosing individual cells from their
boundary.
The steady-state CSD p(m) strongly depends on the
packing fraction η, with more and more cells moving in
larger clusters for increasing packing fraction η. This is
evident in Fig. 4, where we observe that at small values of η, p(m) exhibits a monotonic sharp decay with m,
while at large η values, p(m) is non-monotonic, with an
additional peak at large cluster sizes. The solid curves in
Fig. 4 are fitted to the raw data by using phenomenological functional forms described in the next section. The
CSD here was determined at a fixed time (450 minutes)
after the beginning of each experiment; control experiments at other times (360 minutes, 480 minutes) revealed
practically identical behavior. We interpret the presence
of a peak at large values of m at bigger values of the
packing fractions as the emergence of collective motion
resulting in formation of large clusters of bacteria mov-
ing in a coordinated fashion. The clustering transition
is evident by the functional change displayed by p(m),
monotonically decreasing with m for small values of η,
while exhibiting a local maximum at large η values. At
a critical value ηc = 0.17 ± 0.02 that separates different
regimes of behavior, the CSD can be approximated by
p(m) ∝ m−γ0 , with γ0 = 0.88 ± 0.07. Control experiments with non-motile cells do not exhibit a power-law
behavior in the CSD. For more details, we refer to reader
to [7]. Hence, we conclude that without active motion
of cells no comparable transition to clustering occurs. In
other words, active motion is required for the dynamical
self-assembly of cells.
Now, we turn our attention to the cluster shape, in
particular to the scaling of the cluster perimeter Π(m)
with the cluster size m. This kind of information can
help us to realize how adhesive cells are and which role
adhesion plays in the clustering process. If there is surface tension, then cluster should tend to minimize their
surface, and they should be round, as observed in liquidvapor drops [10]. On the other hand, if surface tension
is negligible, cluster can be very elongated object, with
most of the cells on the cluster boundary, and the cluster perimeter is proportional to cluster size. We assume
that Π(m) ∝ mω , where m denotes the area of the cluster. Thus it is clear that perimeter exponent ω should be
0.5 for round clusters. This would be the case for very
adhesive cells exhibiting random movements. If clusters
are extremely elongated, then ω = 1. We notice that
ω = 1 would correspond also a fully random process as
observed in percolation theory [10]. In short, the exponent ω is then such that 0.5 ≤ ω ≤ 1. Fig. 5 shows that
for A+ S− Frz− cells ω = 0.60 ± 0.03, which indicates that
the clustering process is non trivial that it neither fully
random nor dominated by surface tension, see also Fig.
1. The scaling of Π(m) with m plays a central role in
the clustering theory we discuss below, where the relation between cluster size statistics and cluster perimeter
statistics will be discussed in detail.
As the density increases, typical above η > 0.26, cells
do not organize into large moving clusters, and giant clusters evolve into vortices. These vortices are formed by
one or several layers of rotating disks whose radii dimin-
5
FIG. 3: Time convergence towards a steady state. The figure compares the cumulative cluster size distribution (CCSD), defined
as p(x <= m), at various time points for A+ S− Frz− and A+ S+ Frz− cells at two different packing fractions. The CCSD is less
noisy than the CSD and the comparison at various time points becomes possible. The first row, corresponding to A+ S− Frz−
cells, indicates that the cluster statistics quickly converges to a steady state. The time convergence for A+ S+ Frz− cells, second
row, also occurs, though the phenomenon is less evident. Each panel shows, as reference, the CCSD obtained with control
experiments of non-motile cells. The comparison indicates that cell motility promotes undoubtedly the formation of large
clusters.
FIG. 4: Asymptotic cluster size distribution (t = 450 min) at various packing fractions η for non-reversing mutants A+ S− Frz−
and A+ S+ Frz− and A+ S+ Frz+ . The three strains exhibit a cluster dynamics that evolves towards a steady cluster size
distribution which is function of the cell packing fraction. The cluster size distribution (CSD) for A+ S− Frz− and A+ S+ Frz−
cells exhibits a qualitative change at a critical packing fraction ηc ∼ 0.17; for η > ηc the CSD is no longer monotonically
decreasing distribution and a peak at large cluster sizes emerges. At the critical point, p(m) ∝ m−ξ , with ξ ∼ 0.88. The CSD
of the densities examined can always be approximated by a power-law with an exponential cut-off. Reversing, fully motile
A+ S+ Frz+ cells (wild-type) display an asymptotic CSD which is for all packing fractions η < η ∗ exponential. For η > η ∗ ,
clusters connect such that cells form a mesh-like structure as shown in Fig. 1.
ish the higher the disk is located in the z-direction. Fig. 6
shows a typical example of vortex formation; see the supplementary material for a movie and [59] for a brief description of the movie. Notice that these vortices are not
disordered aggregates of cells as suggested in [25]. Given
the fact that vortices are multilayered structures, phase
contrast imaging can only provide limited information regarding the actual cell arrangements inside vortices. A
detailed study of vortices requires more sophisticated experimental techniques.
6
Interestingly, vortex formation has been also observed
in other experimental “self-propelled rod” systems as
actin-myosin motility assays [26, 35] as well as in 2D
suspensions of sperms [27]. In the later example, hydrodynamical interactions are supposed to induce the
observed pattern, while in the former ones the role of
hydrodynamic interactions is not well understood; yet
in both type of systems the vortex patterns correspond
to vortex arrays. In myxobacteria, on the other hand,
hydrodynamical effects can be neglected and vortices do
not emerge in a lattice-like arrangement, but rather in a
disorganized fashion. At a theoretical level, vortices has
been found in active gel theory [57, 58]. Wether active gel
vortices and those observed in M. xanthus mutants have
the same microscopic origin is unclear, but certainly a
possibility worth exploring.
In summary, the finding of vortex formation in experiments with A+ S− Frz− indicates that the S-motility system, cell-to-cell signaling, and cell reversals are not required for the organization of cells into vortices.
B.
A- and S-motile, non-reversing cells
We turn our attention to the next simplest mutant:
A+ S+ Frz− . These cells contain both motility engines
found in the wild-type, while cell-reversals are absent.
The S-motility system depends on type IV pili [17]. It
allows cells to move in a contact-dependent manner, i.e.
cells have to be in close proximity for S-motility to become active. As previously reported [17], we find that
A+ S+ Frz− cells are more adhesive. Our aim is to understand whether the S-motility engine affects the spatial
self-organization of cells. We performed the same analysis on A+ S+ Frz− cells as described for A+ S− Frz− cells
and investigate cell densities close to the obtained critical density. Fig. 1 shows that at least at first glance the
cluster statistics resembles that obtained with A+ S− Frz−
cells. This suggests that the additional motility including its adhesion effect has no significant impact on the
organization of cells within a cluster. By looking in more
detail on the clustering data some subtle differences can
be revealed. We observe that for all fixed packing fractions η, the CSD seems to evolve towards a steady state,
Fig. 3. However, the temporal convergence is slower than
the one observed for A+ S− Frz− cells. Assuming that
CSD after 450 min from the beginning of the experiment
is representative of the steady state CSD, we show in Fig.
4 the asymptotic behavior of the CSD with packing fraction η. The CSDs of the packing fractions η < 0.18 can
be roughly approximated by a power-law, p(m) ∝ m−γ0 ,
with a critical exponent γ0 consistent with the one obtained for A+ S− Frz− cells, i.e., 0.81 ≤ γ0 ≤ 0.95, see
Fig. 4. On the other hand, the data indicates that a local maximum, as the one described above for A+ S− Frz− ,
emerge for η ≥ 0.18, Fig. 4. On the other hand, the cluster shape statistics shows that the scaling of the perimeter Π with the cluster mass m is again consistent with the
one obtained for A+ S+ Frz− cells with ω = 0.62 ± 0.03,
see Fig. 5. Finally, at sufficiently high densities, these
cells also self-organize into vortices.
C.
Wild-type and the effect of cell-reversal
We applied the same analysis to the reversing
A+ S+ Frz+ cells that move by means of both motility
systems.
Fig. 1 shows that the spatial organization of wild-type
cells is dramatically different from the one observed in the
two mutants. Undoubtedly, cell-reversal has a strong impact on the macroscopic behavior of the colony. The CSD
distribution after 450 min is exponential for all η < 0.20
as shown in Fig. 4. The net distance of cell movement is reduced due to cell reversals and cells can only
form small clusters. On the other side, clusters exhibit
a more elongated shape than those found in experiments
with A+ S− Frz− and A+ S+ Frz− cells, as confirmed by
the scaling of the perimeter Π(m) which is characterized
by a very different exponent ω = 0.82 ± 0.03, see Fig.
5. The initial monodisperse phase, characterized by an
exponential CSD and very elongated clusters, undergoes
a transition at packing fractions larger than 0.26. The
new arrangement of cells percolates and the cells organize
into a mesh-like structure, as shown in Fig. 1.
II.
KINETIC MODEL FOR THE CLUSTER
STATISTICS
In the following, we outline a generalized kinetic model
for the cluster-size distribution and compare it to the
above experimental results. In particular, we want to
relate the cluster size distribution data and the cluster
shape statistics. The model equations are built on the
well-established coagulation theory for colloidal particles
originally suggested by Smoluchowski [31], for an early
review see also [32]. A similar phase transition (albeit
with different exponents for the cluster-size distribution
at criticality) was recently obtained in in a model for
reversible polymerization representing a different generalization of the Smoluchowski model [33].
The model studied consists of a system of kinetic equations for the dynamics of the number nj (t) of clusters of
size j at time t. It was first proposed in [11] to describe
clustering in simulations of self-propelled rods. The individual cluster dynamics [? ], as well as the cluster-cluster
dynamics [11, 34] are strongly simplified in this kinetic
theory where the time evolution of the number nj (t) of
7
FIG. 5: Cluster perimeter π(m) as function of the cluster size m for three bacterial strains. The A+ S− Frz− and A+ S+ Frz−
mutants exhibit roughly the same scaling π(m) ∝ mω , with ω ∼ 0.62 for the A+ S− Frz− , ω ∼ 0.60 for the A+ S+ Frz− , suggesting
that an increase in adhesion does not have a strong impact on the cluster shape. On the other hand, cell reversals lead to much
more elongated clusters, as the scaling of the A+ S+ Frz+ cells indicates, with ω ∼ 0.82.
We have assumed that aggregation of cells occurs only
due to cluster-cluster collisions. Following earlier work
[11, 34], the collision rate between clusters of mass j and
k is defined by:
√ v0 σ 0 p
Aj,k =
j+ k ,
(3)
δ
FIG. 6: Vortex formation (circular pattern in the center)
in the non-reversing mutant A+ S− Frz− at high cell density
(η > 0.26). The pattern consists of rotating stacked discs of
cells. These structure are observed in both, A+ S− Frz− and
A+ S+ Frz− strains.
Bj =
clusters of size j is simply given by:
ṅ1 = 2B2 n2 +
N
X
Bk nk −
k=3
N
−1
X
Ak,1 nk n1
k=1
ṅj = Bj+1 nj+1 − Bj nj −
N
−j
X
Ak,j nk nj
k=1
j−1
1X
+
Ak,j−k nk nj−k
2
for j = 2, ....., N − 1
k=1
ṅN = −BN nN +
N −1
1 X
Ak,N −k nk nN −k ,
2
(1)
k=1
where the dot denotes a time derivative and N is the
total number of cells in the system. The cluster-size distribution is then simply obtained from
p(m, t) =
m nm (t)
.
N
where v0 represents the average speed of individual cells,
σ0 is the average scattering cross section
a single√cell
√ of √
which is assumed to be σ0 ≈ L + W = a( κ + 1/ κ)
and δ is the total area of the system. Eq. (3) assumes that cluster have a well-defined direction of motion, which means that the equation is not adequate to
describe cluster-cluster coagulation in experiments with
wild-type cells. This process competes with cluster fragmentation stemming from the escape of individual single
cells from the cluster boundary. The fragmentation rate
is given by the expression
(2)
v0 j ω
v0 j ω
√ ,
=
R0 L
R0 aκ
(4)
where R0 is a proportionality constant that is the only
free parameter in the theory that is used to fix the critical
value ηc ∝ R0−1 at the same values as in the experiment.
The exponent ω in the fragmentation rate has an important role: it represents the scaling between the cluster mass m and the cluster perimeter Π, i.e., Π ∝ mω .
If one assumes large clusters of approximately circular
shape, then ω = 1/2; this special case has been previously studied in [11]. If instead one considers that cells
form elongated narrow clusters, where practically all cells
are near the boundary, then a choice of ω = 1 is appropriate. In practice, the value of ω will depend on the
number j of particles in the respective cluster. For simplicity, we study only the limiting cases ω = 1/2 and
ω = 1 and compare the resulting cluster-size distribution
to the experimental findings. According to the model,
the exponent γ only depends on the scaling of Π(m),
i.e., the exponent ω, while the critical packing fraction
8
ηc is a non-universal quantity. The analysis of Eqs. (1),
performed by direct numerical integration using a fourthorder Runge-Kutta method, reveals that for η ≤ ηc , the
scaling of p(m) takes the form:
p(m) ∝ m−γ0 exp(−m/m0 ) ,
(5)
while above it, i.e., for η > ηc , the scaling is:
p(m) ∝ m−γ1 exp(−m/m1 ) + Cmγ2 exp(−m/m2 ) , (6)
with γ1 , γ2 , m1 , m2 and C constants that depend on η
and system size. Eq. (5) is the result of a system size
study of Eqs. (1) at the critical point (not shown), while
Eq. (6) is just an educated guess. Eqs. (5) and (6) have
been used to fit the experimental data for the cluster
size distributions in the different strains of myxobacteria
shown in Fig. 4. For η < ηc , for either A+ S− Frz− and
A+ S+ Frz− cells we find using Eq. (5) γ0 ∈ [0.80, 0.95] and
m0 ∈ [20, 1300] (m0 ∼ 20 for η = 0.04 and m0 ∼ 1300 for
η = 0.16). Nevertheless, the critical exponent γ0 has been
estimated by the method explained in the Material and
methods section, where γ0 is found to be γ0 = 0.88±0.07.
For wild-type cells, the distribution is strongly dominated
by an exponential tail. Using Eq. (5) we find γ0 ∈ [0, 0.63]
and m0 ∈ [10, 120].
Through Eq. (1), it can be shown that m0 is a function of η that increases as ηc is approached from below as
observed in Fig. 4. According to the kinetic model, the
critical packing fraction ηc is defined by p(m) ∝ m−γ0 at
η = ηc as long as m is below the total number of cells N
in the system. In contrast, for η < ηc the function p(m)
clearly exhibits an exponentially decaying tail at larger
m, as observed in the experiments with A+ S− Frz− and
A+ S+ Frz− cells, Fig. 7. The theoretical CSD p(m, t) was
obtained by numerical integration from an initial condition with n1 = N and ni = 0 for i ≥ 2. The values of the
variables ni of Eq. (1) reached constant steady values after sufficiently large integration times. The steady state
p(m) was found to depend only on the packing fraction
η for a given perimeter scaling characterized by ω. For
both values of ω studied, we find a transition from an
exponentially decaying CSD, described by Eq. (5) for
low densities, to a non-monotonic CSD, described by Eq.
(6), consisting of a power-law behavior for small cluster
sizes and a peak, local maximum, at large cluster sizes,
see Fig. 7. Upon closer inspection of the model results,
one recovers distinctly different exponents for the different model assumptions regarding ω: γ0 = 1.3 for ω = 1/2
and γ0 = 0.85 for ω = 1. Both choices of ω give reasonable qualitative agreement with the experimental data
shown in Fig. 4 above, see Fig 7. Moreover, we find
that the exponent of the cluster-size distribution is nonuniversal and depends sensitively on the choice of the
fragmentation rate in Eq. (4). We expect that changes
in the collision rate for the cluster will have a similarly
strong effect, as discussed below.
The clustering model given by Eq. (1) allows to study
the relationship between the perimeter scaling (charac-
terized by an exponent ω) and the cluster size distribution exponent γ0 . Eq. (1) also predicts the existence
of two CSDs, depending on the the packing fraction η,
i.e., Eqs. (5) and (6). These two predicted distributions
are found in experiments with A+ S− Frz− and A+ S+ Frz−
cells. For the wild type cells, only the CSD given by Eq.
(5) is found. In this context, it is interesting to note that
the results shown in Fig. 7 imply that for ω = 1 one
needs to assume a much lower fragmentation rate - indicated by a much larger value of the parameter R0 than
for ω = 0.5 to obtain the same critical ηc . Beyond the
apparent agreement between the CSD exhibited by Eq.
(1) and the experimentally obtained CSDs for A+ S− Frz−
and A+ S+ Frz− cells, there are important differences. To
obtain a critical exponent γ0 close to 0.88, ω has to be
large, specifically, close to 1, while the experimental measurements on Π(m) revealed ω ∼ 0.6. There are several
possibilities that could explain this discrepancy. For instance, the assumption that the cluster-cluster coagulation is proportional to square root of the cluster mass has
to be revised. An estimation of the scaling of the effective scattering cross section of a cluster with its mass, as
well as an accurate measurement of the functional dependency of cluster speed with cluster mass would allow us
to determine the correctness of Eq. (1). Unfortunately,
such measurements are extremely difficult to obtained.
Nevertheless, the apparent discrepancy suggests that a
possible generalization of the presented clustering theory
would include a modification of Eq. (3).
III.
DISCUSSION
In order to identify the role of the two motility machineries as well as cell reversal machinery on the spacial collective dynamics of M. xanthus, we have analyzed
three bacterial strains: i) a mutant that moves unidirectionally without reversing by the A-motility system
only, ii) a unidirectional mutant that is also equipped
with the S-motility system, and iii) the wild-type that
is equipped with the two motility systems and occasionally reverses its direction of movement. The study of
the two non-reversing mutants revealed the same phenomenology. At low and intermediate densities, nonreversing cells displays collective motion in the form of
large moving clusters, with a critical density above which
clusters can be arbitrarily large. At the critical density,
the two non-reversing strains exhibit a cluster size distribution characterized by roughly the same critical exponent γ0 ∼ 0.88. Even though the two-engine strain is
supposed to be more adhesive than the single A-engine
strain, we found a similar non-trivial scaling of cluster
perimeter with cluster size characterized by an exponent
ω ∼ 0.6. This finding indirectly shows that the clustering process is, for both stains, neither fully random
nor an equilibrium one as in liquid-vapor drops [10]. In
order to connect the statistics on cluster size and cluster shape, we derived a Smoluchowski-coagulation theory
9
FIG. 7: Theoretical predictions for the cluster size distribution (CSD). The cluster size distribution from the kinetic model
depends on the scaling of the fragmentation Bm ∝ m−ω (see text). The figure shows the results for the two limiting cases, in
the left panel ω = 0.5 (and R0 = 0.58) corresponding to round clusters, while in right panel ω = 1 (and R0 = 11.9), implying
elongated clusters. Notice that in theory, as well as in experiments, the CSD can be well approximated by a power-law at the
critical packing fraction ηc , while for η > ηc a peak at large cluster sizes emerges, a signature of dynamic self-assembly into
larger moving clusters. Other model parameters: κ = 9, a = 4.4µm2 , and δ = 699 × 522µm2
with fragmentation, where we related the scaling of cluster perimeter with cluster size with the fragmentation
kernel. The proposed theory allows us to understand
the cluster formation process in absence of adhesion as a
dynamic self-assembly process. It predicts the existence
of a steady state cluster size distribution which is function of the cell density and perimeter exponent ω, and a
functional change of the cluster size distribution above a
critical density. In addition, the proposed theory predicts
that the critical exponent γ0 depends on the perimeter
exponent ω only. In summary, the theoretical clustering
model provides a qualitative descriptions consistent with
the experimental measurements, and explains why if the
value of ω is similar for both strains, the value of γ0 has to
be also similar. We observe that similar spatial organization has been observed in self-propelled rod simulations
using either rigid [11] or elastic [28] elongated particles.
We found that at high densities the collective dynamics changes and cells organize into vortices. This finding
cannot be account by the proposed clustering theory, but
it is reminiscent of what is observed in self-propelled rod
simulations at high densities, though in experiments vortices seem to be stable structures while in simulations
vortices are unstable. From the comparison of these two
non-reversing strains, we conclude that unidirectional cell
motion induces the formation of large moving clusters at
low and intermediate densities, while it results into vortex formation at very high densities. On the light of
the clustering theory and given the remarkable similarity with self-propelled rod simulations, we suggest that
the spatial self-organization in these two strains occurs
in absence of biochemical signal regulations and as result
of the the combined effect of self-propulsion and volume
exclusion interactions. All these results strongly suggest
that the combination of self-propulsion and steric interaction is a valid pattern formation mechanism which could
be also at play in recent experiments with Escherichia
coli [45] and driven actin filaments [35], which makes
us wonder about the connection between this mechanism
and the large body of work on simple models of selfpropelled particles where spontaneous segregation and
long-range orientational has been reported [36–44].
The study of wild-type cells has revealed that cell reversal affects dramatically the collective dynamics. We
found that wild-type cells exhibit cluster size distributions exponentially distributed at low and intermediate
densities. On the other hand, we measured an the scaling
of the cluster perimeter with cluster size characterized by
a large exponent ω ∼ 0.8 which indicates that clusters are
strongly elongated with comparison to those found in experiments with the two non-reversing mutants. Finally,
we observed that at high densities cells self-organize into
a mesh-like structure. A qualitative understanding of this
macroscopic behavior is still missing. The comparison of
the two non-reversing strains and wild-type cells suggest
suggests that by only switching on and off the reversal,
cells can modify dramatically their collective behavior,
with the suppression of reversal leading to collective motion in the form of moving clusters and vortex formation
at high densities. We note that this observation is consistent with the observed decrease in reversal frequency in
the wild-type upon nutrient depletion, which is followed
by the formation of large moving clusters and aggregation
of cells.
At a more speculative level, our results suggest that
the cell density and the rod shape of the cells may play
an essential role for bacteria to achieve collective motion [46, 47]. According to self-propelled rod simulations,
an elongated cell shape strongly facilitates collective motion by promoting the formation of larger clusters. Another hint that the rod-shape of the moving bacteria is
important for collective motion is provided by the empirical observation that many bacteria undergo a dramatic elongation of their cell shape before assembling
10
into larger groups, e.g. in Vibrio parahaemolyticus [48]
or B. subtilis [49]. Finally, the reported results increase
the plausibility of earlier biological hypotheses [46], that
multicellular organization may be achieved by regulating
the cell density via proliferation and cell length by direct
developmental control.
We acknowledge support by the German Ministry for
Education and Research (BMBF) through grants no.
0315259 and no. 0315734, by the Human Frontier Science Program (HFSP) through grant no. RGP0016-2010,
by DFG through grant DE842/2 and the Max Planck
Society. Partial DFG support by SFB 555 and GRK
1558 is also acknowledged. AD is a member of the DFG
Research Center for Regenerative Therapies Dresden –
Cluster of Excellence – and gratefully acknowledges support by the Center. FP acknowledges support by PEPS
PTI “Anomalous fluctuations in the collective motion of
self-propelled particles”.
IV.
MATERIAL AND METHODS
A.
Bacterial strains.
The fully motile strain DK1622 (A+ S+ Frz+ ) was used
as a wild type [50] and all other strains used are derivatives of DK1622. The non-reversing strain DK8505 [51] is
referred to as A+ S+ Frz− . To generate SA2407 cells, here
referred to as A+ S− Frz− , the frz loss-of-function allele
f rzCD::T n5 lac Ω536 from DK8505 [51] was introduced
into the ∆pilA strain DK10410 [52], which is unable to
assemble type IV pili, using standard procedures [53].
To generate SA2082 (∆pilA, romR::nptII), the nonmotile M. xanthus mutant referred to as A− S− Frz− , the
romR::nptII loss-of-function allele from SA1128 [54] was
introduced into DK10410. All strains used had a doubling time of approximately 5 hrs in CTT liquid medium
at 32◦ C. Notice that the relaxation time of spatial patterns is below 120 min which implies that the doubling
time has a weak effect on the spatial patterns.
B.
Cluster formation experiments
Cultures of M. xanthus were grown in CTT liquid
medium [55] at 32o C with shaking to an estimated density of 7 × 109 cells/ml. Subsequently, cells were diluted
to densities of 0.5 × 108 /ml, 1.0 × 108 /ml, 1.5 × 108 /ml,
1.75 × 108 /ml and 2 × 108 /ml, respectively. Cell densities were confirmed by colony counts on CTT agar plates
manually and by counting the number of cells using a
counting chamber manually. 30µl aliquots of cells were
transferred to a microscope slide covered with a 1.0%
agar pad in 0.5% CTT medium. The time point at which
the cell drop was completely absorbed in the agar was set
as t = 0. For each cell density, 16 slides were prepared
and every 30min (starting at 30min) up to 480min, a sample was analysed by microscopy using a Leica DM6000B
microscope with a Leica 20× phase-contrast objective
and imaged with a Leica DFC 350FX camera. 20 phasecontrast images were taken at 20× magnification across
a spot. After 480min a short time-lapse movie was taken
to verify that cells and clusters were migrating.
C.
Image analysis
Clustering images were taken at 20× magnification.
Images contain cell clusters as dark regions, often surrounded by a light halo. Cluster boundaries were detected in a multi-step processing queue. After initial image normalization, edge detection via the Canny-Deriche
algorithm was applied for two different levels of spatial detail. Both edge images were superimposed subsequently. Next, edges were filtered out that surround
halos and other non-cluster objects. Finally, all incomplete detections were revised/corrected manually in
a post-processing step. The areas of the clusters in
pixels were extracted using an implementation of the
processing queue in the image processing tool ImageJ
(http://rsbweb.nih.gov/ij/). The number of cells inside
a cluster, i.e., the cluster size, was estimated as the area
of a cluster divided by the mean area covered by a single cell, which was found to be 150 pixels at 20-fold
magnification. According to this definition, a cluster
is a connected group of cells, regardless of their orientation. Packing fraction estimates per image were obtained as the ratio of area covered by cells and the whole
area of the image (1392 × 1040 pixels corresponding to
699µm × 522µm).
D.
Statistical analysis
After applying the image analysis procedure described
above to a given image I, corresponding to a given packing fraction, a large array of various cluster sizes is obtained, and nI (m, t) can be computed. We represent by
nI (m, t) the number of clusters of size m in the image I.
To build the CSD we make use of all the available images
corresponding to the given packing fraction η. Let the
auxiliary function gI (m, t) be gI (m, t) = m nI (m, t). The
average of this function reads:
g(m, t) = (1/M )
X
gI (m, t) ,
(7)
I
where M is the number of available images. To cope with
the sparseness of the data for large cluster sizes we implemented several binning procedures, in particular, linear
and exponential binning. In the following we explain the
exponential binning procedure. The cluster size space is
divided into bins, the first bin contains all clusters of size
s, 0 < s ≤ 1, the second bin all clusters of size 1 < s ≤ 2,
the third bin, 2 < s ≤ 4,... the n-th bin contains cluster
11
FIG. 8: Procedure to estimate the critical exponent γ0 and critical packing fraction ηc . a) The standard deviation of the
transformation given by Eq. (10) with respect to its mean value W , see Eq. (11). The shown data corresponds to A+ S− Frz−
cells. The minimum exhibited by η = 0.16 − 0.18 indicates that the transformation given by Eq. (10) leads to a horizontal line,
indirectly showing that the values of η can be well fitted by Eq. (5). b) shows the sensitivity of this procedure. If the critical
exponent is either overestimated or underestimated, there is no transformed CSD, for any value of η, yielding a horizontal line.
Only very close to the actual critical exponent γ0 , the transformation can be approximated by a constant W .
sizes 2n−1 < s ≤ 2n . It is useful to define the function:
following transformation yields to a constant:
e(n+1)
gbin (n, t) =
X X
e(n)
gI (m, t)
(8)
y(m) = p(m; ηc )mγ = W
(9)
where W is a constant and the equality holds true for
1 < x < xcut−of f , where xcut−of f denotes the beginning
of the cut-off. The value of W is the average value of
y(m) in the interval 1 < x < xcut−of f . This means that
if we plot y(m) vs. m, we observe a horizontal line at the
critical packing fraction ηc . We can measure how close
we are to the horizontal line by computing:
I
where e(n) = 2n . The binned CSD is defined as:
pbin (e(n), t) =
(10)
gbin (n, t)
.
C(e(n − 1) − e(n))
P
Thus,
m pbin (m = e(n), t)(e(n − 1) − e(n)) = 1. It
is worth noticing that if the underlying CSD p(m) is a
power-law characterized by an exponent γ, i.e., p(m) ∼
m−γ , the exponential binning procedure given by Eq.
(9) results in pbin (m) ∼ m−γ . On the other hand, if
the underlying CSD p(m) is an exponential, i.e., p(m) ∼
exp(m/m0 ), the exponential binning leads to pbin (m) ∼
m−1 exp(m/m0 ). In the text, for simplicity we referred
to pbin (m, t) just as p(m, t).
In what follows, we explain how the critical exponent
has been measured. At the critical packing fraction ηc
the CSD is a power-law (with an exponential cut-off due
to the finite number of cells). The problem consists in
identifying the critical packing fraction ηc and the critical exponent γ0 . Assuming that we know γ0 at ηc the
[1] Bretschneider T, Siegert F, Weijer CK (1995) Threedimensional scroll waves of cAMP could direct cell movement and gene expression in Dictyostelium slugs. Proc.
Natl. Acad. Sci. USA 92:4387-4391.
[2] Ben-Jacob E, Cohen I, Levine H (2000) Cooperative self-
σ 2 (η, γ) =
X
(y(m) − W )2
(11)
By minimizing Eq. (11) with respect to η and γ, the
critical packing fraction and critical exponent can be obtained. Figure 8 illustrates the procedure. In the figure
the cut-off was taken to xcut−of f = 220 (various other
values were also studied). We found that the critical
packing fraction lays between 0.16 and 0.18 for either
A+ S− Frz− or A+ S+ Frz− cells and the critical exponent
is γ0 = 0.88 ± 0.07 and γ0 = 0.85 ± 0.07 for A+ S− Frz−
or A+ S+ Frz− cells, respectively.
organization of microorganisms. Adv. Phys. 49:395-554.
[3] Kaiser D (2003) Coupling cell movement to multicellular
development in myxobacteria. Nat. Rev. Microbiol. 1:4554.
[4] Dombrowski C, Cisneros L, Chatkaew S, Goldstein
12
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
RE, Kessler JO (2004) Self-concentration and largescale coherence in bacterial dynamics. Phys. Rev. Lett.
93:098103.
Zhang HP, Be’er A, Florin E-L, Swinney HL (2010)
Collective motion and density fluctuations in bacterial
colonies. Proc. Natl. Acad. Sci. USA 107:13626-13630.
M. E. Cates et al. (2012). Arrested phase separation in
reproducing bacteria: a generic route to pattern formation?. Proc. Natl. Acad. Sci. U.S.A. 107, 11715-11720.
F. Peruani, J. Starruß, V. Jakovljevič, L. SøgaardAndersen, A. Deutsch, and M. Bär (2012) Collective motion and nonequilibrium cluster formation in colonies of
gliding bacteria, Phys. Rev. Lett., 108, 098102.
Whitesides GM, Grzybowski B (2002) Self-assembly at
all scales. Science 295:2418–2421.
Copeland MF, Weibel DB (2009) Bacterial swarming: a
model for studying dynamic self-assembly. Soft Matter
5:1174-1187.
D. Stauffer (1979), Scaling theory of percolation clusters,
Phys. Rep 54, 1-74 .
Peruani F, Deutsch A, Bär M (2006) Nonequilibrium
clustering of self-propelled rods. Phys. Rev. E 74:030904030908(R).
Yang Y, Marceau V, Gompper G (2010) Swarm behavior
of self-propelled rods and swimming flagella. Phys. Rev.
E 82:031904.
Kudrolli A, Lumay G, Volfson D, Tsimring LS (2008)
Swarming and swirling in self-propelled granular rods.
Phys. Rev. Lett. 100:058001.
Stevens A, Søgaard-Andersen L (2005) Making waves:
Pattern formation by a cell surface-associated signal.
Trends Microbiol. 13:249-252.
Vos M, Velicer GJ (2006) Genetic population structure of
the soil bacterium Myxococcus xanthus at the centimeter
scale. Appl. Environ. Microbiol. 72:3615-3625.
Mauriello EM, Mignot T, Yang Z, Zusman DR (2010)
Gliding motility revisited: how do the myxobacteria
move without flagella?. Microbiol. Mol. Biol. Rev. 74:229249.
Wu SS, Kaiser D (1995) Genetic and functional evidence
that Type IV pili are required for social gliding motility
in Myxococcus xanthus. Mol. Microbiol. 18:547-558.
Hodgkin J, Kaiser D (1979) Genetics of gliding motility
in Myxococcus xanthus: two gene systemss control movement. Mol. Gen. Genet. 171:177-191.
Blackhart BD, Zusman DR (1985) Frizzy genes of Myxococcus xanthus are involved in control of frequency of
reveral of gliding motility. Proc. Natl. Acad. Sci. USA
82:8771-8774.
Jelsbak L, Søgaard-Andersen L. (2002) Pattern formation by a cell surface associated morphogen in M. xanthus. Proc. Natl. Acad. Sci. USA 99:2032-2037.
O’Connor KA, Zusman DR (1989) Patterns of cellular interactions during fruiting-body formation in Myxococcus
xanthus. J. Bacteriol. 171:6013-6024.
Wolgemuth C, Hoiczyk E, Kaiser D, Oster G (2002) How
myxobacteria glide. Curr. Biol. 12:369–377.
Mignot T, Shaevitz JW, Hartzell PL, Zusman DR (2007)
Evidence of focal adhesion complexes power bacterial
gliding motility. Science 315:853-856.
Pelling AE, Li Y, Cross SE, Castaneda S, Shi W,
Gimzewski KJ (2006) Self-organized and highly ordered
domain structures within swarms of Myxococcus xanthus,
Cell Motil. Cytoskeleton 63: 141–148.
[25] Holmes AB, Kalvala S, Whitworth DE (2010) Spatial
simulations of myxobacterial development. PLoS Comput Biol 6(2): e1000686.
[26] Sumino Y. et al. (2012) Large-scale vortex lattice emerging from collectively moving microtubules. Nature 483,
448-52
[27] Riedel I., Kruse K., Howard J. (2007) A Self-Organized
Vortex Array of Hydrodynamically Entrained Sperm
Cells. Science 309, 300-303
[28] StarrußJ, Bley T, Søgaard-Anderson L, Deutsch A (2007)
A new mechanism for collective migration in Myxococcus
xanthus. J. Stat. Phys. 128:269–286.
[29] Stevens A (2000) A stochastic cellular automaton, modeling gliding and aggregation of myxobacteria. SIAM J.
Appl. Math. 61:172-182.
[30] Sliusarenko O, Zusman DR, Oster G (2007) Aggregation
during fruiting body formation in Myxococcus xanthus is
driven by reducing cell movement. J. Bacteriol. 189:611–
619.
[31] von Smoluchoswki M (1917) Versuch einer mathematischen Theorie der Koagulationskinetik kolloider
Lösungen. Z. Phys. Chem. 92:129.
[32] Chandrasekhar S (1943) Stochastic problems in physics
and astronomy. Rev. Mov. Phys. 15:1–89.
[33] Ben-Naim E, Krapivsky PL (2008) Phase transition with
nonthermodynamic states in reversible polymerization.
Phys. Rev. E 77:061132.
[34] Peruani F., Schimansky-Geier L., and Bär M. (2010)
Cluster dynamics and cluster size distribution in systems
of self-propelled particles, Eur. Phys. J. Special Topics
191, 173-185.
[35] Schaller V, Weber C, Semmrich C, Frey E, Bausch AR
(2010) Polar patterns of driven filaments. Nature 467:73–
77.
[36] Vicsek T, Czirok A, Ben-Jacob E, Cohen I, Shochet O
(1995) Novel type of phase transition in a system of selfdriven particles. Phys. Rev. Lett. 75:1226.
[37] Toner J, Yu Y (1995) Long-range ordder in a twodimensional XY model: How birds fly together. Phys.
Rev. Lett. 75:4326.
[38] Bussemaker HJ, Deutsch A, Geigant E (1997) Mean-field
analysis of a dynamical phase transition in a cellular automaton model for collective motion. Phys. Rev. Lett.
78:5018.
[39] Grégoire G., Chaté H. (2004) Onset of collective and cohesive motion. Phys. Rev. Lett. 92:025701.
[40] Mishra S, Ramaswamy S (2006) Active nematics are intrinsically phase separated. Phys. Rev. Lett. 97:090602.
[41] Ramaswamy S, Aditi Simha R, Toner J (2003) Active
nematics on a substrate: giant number fluctuations and
long-time tails. Europhys. Lett. 62:196.
[42] Chaté H, Ginelli F, Grégoire G, Raynaud F (2008) Collective motion of self-propelled particles interacting without
cohesion. Phys. Rev. E 77:046113.
[43] Baskaran A, Marchetti MC (2008) Hydrodynamics of
self-propelled hard rods. Phys. Rev. E 77:011920.
[44] Ginelli F, Peruani F, Bär M, Chaté H (2010) Large-scale
collective properties of self-propelled rods. Phys. Rev.
Lett. 104:184502.
[45] Drescher K, Dunkel J, Cisneros LH, Ganguly S, Goldstein
RE (2011) Fluid dynamics and noise in bacterial cell-cell
and cell-surface scattering. Proc. Natl. Acad. Sci. USA
108:10940–10945
[46] Young KD (2006) The selective value of bacterial shape.
13
Microbiol. Mol. Biol. Rev. 70:660–703.
[47] Harshey RM (1994) Bees aren’t the only ones: swarming
in Gram-negative bacteria. Mol. Microbiol. 13:389-394.
[48] McCarter L (1999) The multiple identities of Vibrio parahaemolyticus. J. Mol. Microbiol. Biotechnol. 1:51–58.
[49] Julkowska D, Obuchowski M, Holland IB, Seror SJ (2004)
Branched swarming patterns on a synthetic medium
formed by wild-type Bacillus subtilis strain 3610: detection of different cellular morphologies and constellations
of cells as the complex architecture develops. Microbiology 150:1839-1849.
[50] Kaiser D (1979) Social gliding is correlated with the presence of pili in Myxococcus xanthus. Proc. Natl. Acad. Sci.
USA 76:5952.
[51] Sager B, Kaiser D (1994) Intercellular C-signaling and
the traveling waves of Myxococcus. Genes Dev. 8:2793.
[52] Wu SS, Kaiser D (1997) Regulation of expression of
the pilA gene in Myxococcus xanthus. J. Bacteriol.
179(24):7748.
[53] Higgs PI et al. (2005) Four unusual two-component signal
transduction homologs, RedC to RedF, are necessary for
timely development in Myxococcus xanthus. J. Bacteriol.
187:8191.
[54] Leonardy S, Freymark G, Hebener S, Ellehauge E,
Søgaard-Andersen Lotte (2007) Coupling of protein localization and cell movements by a dynamically localized response regulator in Myxococcus xanthus. EMBO
J. 26:4433.
[55] Hodgkin J, Kaiser D (1977) Cell-to-cell stimulation of
movement in non-motile mutants of Myxococcus. Proc.
Natl. Acad. Sci. USA 74:2938.
[56] Hodgkin J, Kaiser D (1979) Genetics of gliding motility
in Myxococcus xanthus (Myxobacterales): Two gene systems control movement. Mol. Gen. Genet. 171:547–558.
[57] J. Elgeti, M. E. Cates and D. Marenduzzo (2011) Soft
Matter 7, 3177.
[58] Kruse K., Joanny J. F., Julicher F., Prost J., and Sekimoto K., Asters (2004),Vortices, and Rotating Spirals
in Active Gels of Polar Filaments, Phys. Rev. Lett. 92,
078101.
[59] The movie shows a vortex of A+S-Frz- cells, magnification 20×, real time duration 15 min.